PSI of water in a cup

[Idea04]

Homework Statement

If you take a cup of water upside down on top of water in a bowl. The water doesn't flow out because of atmospheric force. So my question is, is the atmospheric pressure of 14.7psi transferred to the water in the cup. So is the water pressure inside the cup then 14.7 psi?

Homework Equations

The Attempt at a Solution

 

[mybsaccownt]

if the pressure inside the cup was not equal to the atmospheric pressure, what would happen?

 

[Idea04]

So water open to the air will always have a pressure of 14.7psi. plus its own pressure.

 

[Idea04]

So if the atmospheric pressure is transferred to the water, If you take a column of water 12 feet vertical with the top closed off and water entering at the bottom, with no air inside the column the water would rise to even pressure. So my question is would the atmospheric pressure be transferred throughout the water in the column. So at the top of the column would the pressure be 14.7 psi?

 

[mybsaccownt]

well, think about the units for stating a pressure as a height of a fluid

mmHg (milimeters of mercury)

760 mmHg = 14.7psi


that is, a column of mercury 760mm high will exert a pressure at the bottom, equal to atmospheric pressure


note that mercury is much denser than water and the water column that would provide a pressure of 14.7 psi would actually be 13.6 times taller

(because the density of water is 13.6 times that of mercury)


the pressure at the bottom of a column of liquid is due to its weight

it can be calculated from:

P = rho*g*h

rho being the density of the fluid, g being gravity, and h being the height of the column

 

[mybsaccownt]

So at the top of the column would the pressure be 14.7 psi?

err

at the top of the column...

no

the pressure that the column of water exerts is the weight of all the fluid

so at the top of the column it's just vapor pressure which is negligibly small

hmm...let me think about it a little more

i think it depends on the size of the glass

a small glass would not have a water column that would provide enough pressure to balance with the atmospheric pressure, so it seems that the pressure at the top would be the pressure needed to reach atmospheric after taking into account the pressure provided from the small column of water


alternatively...a glass that's 15 m tall, would be able to balance atmospheric pressure with the height of the water column it can provide, so the pressure at the top of the glass would basically be 0

 

[Idea04]

okay, But having a column of water closed off at the top so no air can push downward is different then water sitting in a open top container.
The pressure of the atmosphere pushes the water up into the column. Keeping in mind that there is no air inside the column. So if the column was filled to the top it would exert around 5 psi downward pressure. That would be less than atmospheric pressure.
So my way of thinking, the pressure would be 14.7 psi in the column.
But would that be transferred to all parts of the column, or would the top of the column be 9.7psi and the bottom be 14.7 psi?

 

[mybsaccownt]

The pressure of the atmosphere pushes the water up into the column. Keeping in mind that there is no air inside the column. So if the column was filled to the top it would exert around 5 psi downward pressure. That would be less than atmospheric pressure.
So my way of thinking, the pressure would be 14.7 psi in the column.
But would that be transferred to all parts of the column, or would the top of the column be 9.7psi and the bottom be 14.7 psi?

if the glass was full, and the water in the bowl is trying to force its way in with atmospheric pressure

then the pressure at the top of the glass is the difference in pressure between atmospheric and the pressure provided by the water column inside the glass

I suppose you could consider the water column to provide negligible pressure and say that the pressure at the top of the glass is atmospheric pressure

it wouldn't be the exact answer, but a water column in a normal sized glass wouldn't provide very much pressure at all

 

[mybsaccownt]

the main thing, is that the pressure where the glass meets the free surface has to be atmospheric pressure

it'll be a constant at any point in that elevation

looking at the surface of the water in the bowl, it has atmospheric pressure at that elevation because that is what is causing the pressure at that point (the weight of the air column above it)

looking at the water at the bottom of the glass at the same elevation as the surface of the water in the bowl, it has atmospheric pressure at this elevation but it is not due to the column of air above it, it is due to the weight of the column of water above it and the pressure inside the glass

since no size of glass was given, it would not be wise to assume the pressure provided by the column of water is negligible

therefore,

the pressure at the top of the glass is the difference in pressure between atmospheric and the pressure provided by the column of water


you can apply bernoulli's equation to the top of the glass and the free surface in order to reach this conclusion as well